Theorem curry1val 8088

Description: The value of a curried function with a constant first argument. (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)

Hypothesis

Ref	Expression
curry1.1	⊢ 𝐺 = (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))

Assertion

Ref	Expression
curry1val	⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (𝐺‘𝐷) = (𝐶𝐹𝐷))

Proof of Theorem curry1val

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		curry1.1	. . . 4 ⊢ 𝐺 = (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))
2	1	curry1 8087	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)))
3	2	fveq1d 6891	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (𝐺‘𝐷) = ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷))
4		eqid 2733	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))
5	4	fvmptndm 7026	. . . . . 6 ⊢ (¬ 𝐷 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = ∅)
6	5	adantl 483	. . . . 5 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ ¬ 𝐷 ∈ 𝐵) → ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = ∅)
7		fndm 6650	. . . . . . 7 ⊢ (𝐹 Fn (𝐴 × 𝐵) → dom 𝐹 = (𝐴 × 𝐵))
8	7	adantr 482	. . . . . 6 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → dom 𝐹 = (𝐴 × 𝐵))
9		simpr 486	. . . . . . 7 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 𝐷 ∈ 𝐵)
10	9	con3i 154	. . . . . 6 ⊢ (¬ 𝐷 ∈ 𝐵 → ¬ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵))
11		ndmovg 7587	. . . . . 6 ⊢ ((dom 𝐹 = (𝐴 × 𝐵) ∧ ¬ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → (𝐶𝐹𝐷) = ∅)
12	8, 10, 11	syl2an 597	. . . . 5 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ ¬ 𝐷 ∈ 𝐵) → (𝐶𝐹𝐷) = ∅)
13	6, 12	eqtr4d 2776	. . . 4 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ ¬ 𝐷 ∈ 𝐵) → ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = (𝐶𝐹𝐷))
14	13	ex 414	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (¬ 𝐷 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = (𝐶𝐹𝐷)))
15		oveq2 7414	. . . 4 ⊢ (𝑥 = 𝐷 → (𝐶𝐹𝑥) = (𝐶𝐹𝐷))
16		ovex 7439	. . . 4 ⊢ (𝐶𝐹𝐷) ∈ V
17	15, 4, 16	fvmpt 6996	. . 3 ⊢ (𝐷 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = (𝐶𝐹𝐷))
18	14, 17	pm2.61d2 181	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = (𝐶𝐹𝐷))
19	3, 18	eqtrd 2773	1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (𝐺‘𝐷) = (𝐶𝐹𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475  ∅c0 4322  {csn 4628   ↦ cmpt 5231   × cxp 5674  ^◡ccnv 5675  dom cdm 5676   ↾ cres 5678   ∘ ccom 5680   Fn wfn 6536  ‘cfv 6541  (class class class)co 7406  2^nd c2nd 7971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-1st 7972  df-2nd 7973

This theorem is referenced by:  nvinvfval  29881  hhssabloilem  30502